refraction and reflexion according to the principle...

REFRACTION AND REFLEXION ACCORDING TOTHE PRINCIPLE OF GENERAL COVARIANCE

PATRICK IGLESIAS-ZEMMOUR

Abstract. We show how the principle of general covariance introduced bySouriau in smoothly uniform contexts, can be extended to singular situ-ations, considering the group of diffeomorphisms preserving the singularlocus. As a proof of concept, we shall see how we get again this way, thelaws of reflection and refraction in geometric optics, applying an extendedgeneral covariance principle to Riemannian metrics, discontinuous alonga hypersurface.

Introduction

In his paper “Modèle de particule à spin dans le champ électromagnétique etgravitationnel” published in 1974 [Sou74], Jean-Marie Souriau suggests a pre-cise mathematical interpretation of the principle of General Relativity. Henames it the Principle of General Covariance. Considering only gravitationfield1, he claimed that any material presence in the universe is characterizedby a covector defined on the quotient of the set of the Pseudo-Riemannianmetrics on space-time, by the group of diffeomorphisms. This principle be-ing, according to Souriau, the correct statement of the Einsteins’s principleof invariance with respect to any change of coordinates. Actually, Souriau’sgeneral covariance principle can be regarded as the active version of Einsteininvariance statement, where change of coordinates are interpreted from theactive point of view as the action of the group of diffeomorphisms.

Now, for reasons relative to the behavior at infinity and results requirement,the group of diffeomorphisms of space-time is reduced to the subgroup ofcompact supported diffeomorphisms.

Date: April 6, 2019.1991 Mathematics Subject Classification. 83C10, 78A05, 37J10.Key words and phrases. General Covariance, Geometric Optics, Symplectic Geometry.1In his paper Souriau extends also his consideration to electromagnetic field.

1

2 PATRICK IGLESIAS-ZEMMOUR

In consequence, after some identifications, Souriau interprets a material pres-ence for a Pseudo-Riemannian metric g on the space-time M, responding tothis principle of covariance, as a tensor distribution T whose test functionsare compacted supported covariant 2-tensors δg defined on M, which van-ishes along the infinitesimal action of the compact supported diffeomorphismsgroup Diffcpct(M). That is, in Souriau’s notations:

T (δL g ) = 0,

where δL g is the Lie derivative of the metric g by any compact supportedvector field δx . In a more standard notation, denoting by ξ the compact sup-ported vector field on M, δL g = £ξ(g ). A tensor distribution satisfying thatlast equation has been called by Souriau, an Eulerian distribution.Among others, there are two main examples of application of this principle.The first one is a continuous distribution of matter described by aC1-smoothsymmetric contravariant 2-tensor Tµν,

T (δg ) =12

∫

MTµνδgµν vol, (1)

where vol is the Riemannian volume associated with g , and we use the Ein-stein’s convention on (up–down) repeated indices. We recall that the Lie de-rivative of a metric g by a vector field ξ is given, in terms of coordinates,by

δL gµ,ν = ∂̂µξν+ ∂̂νξµ,

where ∂̂ denotes the covariant derivative with respect with the Levi-Civitaconnection relative to g (sometimes denoted also by ∇). Then, the Euleriancondition above writes now:

12

∫

MTµν(∂̂µξν+ ∂̂νξµ)vol= 0.

By reason of symmetries and integrating by parts, recalling that ξ is compactsupported, one gets:

∫

M∂̂µ(T

µν)ξνvol= 0,

for all compact supported vector field ξ. Therefore the Eulerian conditionwrites:

∂̂µ(Tµν) = 0, or d̂iv(T) = 0,

where d̂iv(T) is the Riemannian divergence of the tensor T. Hence, the Euler-ian character of the distributionT translates simply in the Euler conservation

REFRACTION AND REFLEXION ACCORDING TO GENERAL COVARIANCE 3

equations of the energy-momentum tensor. Which is the first principle inGeneral Relativity. And that justifies a posteriori the chosen vocabulary.

The second example is a preamble to the subject of our paper. It concerns theconstruction of geodesics as Eulerian distributions supported by a curve. So,let t 7→Tµν be aC2-smooth curve in the fiber bundle of symmetric contravari-ant 2-tensor over M, over a curve γ : t 7→ x . Let then,

T (δg ) =12

∫ +∞

−∞Tµνδgµν dt .

After some astute manipulation [Sou74], Souriau shows that this distributionis Eulerian if and only if the supporting curve γ is a geodesic, that is,

d̂dt

dxdt= 0.

And then:

T (δg ) =k2

∫ +∞

−∞

dxµ

dtdxν

dtδgµν dt ,

where k can be any real constant. In general k is interpreted as the mass mof the particle moving along the geodesic γ with velocity v= dx/dt . But forclassical geometric optics k will be interpreted as the color coefficient hν/c2,where h is the Planck constant, ν is the frequency of the ray and c is the speedof light in vacuum.

Thus, the condition to be geodesic is interpreted through this approach as animmediate consequence of the principle of General Relativity, which is quitesatisfactory.

This Souriau’s construction has been also commented in a great way, and ap-plied unexpectedly by Shlomo Sternbeg, to derive the Schrödinger’s equationfrom the same covariance principle [Ste06].

In this paper, we shall see how, this general covariance principle can also beextended to the case of an interface. That is, a singular situation where themetric is not anymore smooth, but has a discontinuity along a hypersurface.This is interpreted as a refraction/reflexion problem, as we shall see. And weshall get then, as an application of this extended general covariance principle,the Snell–Descartes law of reflection and refraction of light.

Thanks. — I am thankful to the referee for the careful reading of the manu-script, and for his remarks which contributed to enhance the content of thispaper.


The General Covariance Principle with interface

In the following we shall consider the following situation, described by Fig. 1.A manifold M is split in two parts M1 and M2 by an embedded hypersurfaceΣ. We can consider thatM1 andM2 are twomanifolds with a shared boundaryΣ, but their union is the smooth manifold without boundary M:

M1 ∩M2 = Σ and M1 ∪M2 =M.

Next, on each part M1 and M2 we define two smooth Riemannian (or Pseudo-Riemannian) metrics g1 and g2. That means precisely that g1 is the restrictionto M1 of a metric defined on a small open neighborhood of M1, idem for g2with M2. The two metrics have then a limit on Σ which may not coincide.We shall denote by g = (g1, g2) this metric on M, having a discontinuity onΣ.Next, we introduce the subgroup of compact supported diffeomorphisms ofM preserving the hypersurface Σ,

Diffcpct(M,Σ) = {φ ∈Diffcpct(M) | φ(Σ) = Σ}.

Actually we want φ to preserve separately the two parts M1 and M2. That is,φ ∈Diffcpct(M), φ �M1 ∈Diffcpct(M1), φ �M2 ∈Diffcpct(M2), and of courseφ � Σ ∈Diffcpct(Σ). To say that φ preserves Σ and is connected to the identitywould be sufficient.Now, we introduce the test tensors for the distribution tensors relative to thisconfiguration. Let g = (g1, g2) be a metric on M = (M1,M2) as describedabove and let s 7→ gs = (g1,s , g2,s ) be two smooth paths in the set of metrics,pointed at g , that is, g0 = g . We shall denote δg = (δg1,δg2) the variationof gs at s = 0,

δg1 =∂ g1,s

∂ s

�

�

�

�

�

s=0

and δg2 =∂ g2,s

∂ s

�

�

�

�

�

s=0

.

Therefore, an infinitesimal diffeomorphism which preserves the figure gives avariation of g which is a Lie derivative:

δL g = (δL g1,δL g2) with δL gi =∂ (e sξ)∗(gi )∂ s

�

�

�

�

�

s=0

,

where ξ is a smooth compact supported vector field on M such that ξ(x) ∈TxΣ for all x ∈ Σ. Its exponential e sξ belongs to Diffcpct(Σ). We have then

(δL gi )µν = ∂̂i ,µξν+ ∂̂i ,νξµ,


where ∂̂i is the covariant derivative with respect to gi . Let us come back tothe tensor distribution: according to what was said until now, by linearity Tsplits in the sum of two tensors, each relative to a part of M. That is,

T (δg ) =T1(δg1)+T2(δg2). (2)

The tensor distribution T1 is equal to T (δg ) for δg2 = 0, as well for T2 withδg1 = 0. Therefore, for T to be Eulerian means:

T (δL g ) = 0 ⇔ T1(δL g1)+T2(δL g2) = 0. (3)

Crossing the Border

As a first case, let us consider a continuous medium described by a tensor Tµν,extended on M1 and M2. That is,

T (δg ) =12

∫

MTµν

1 δg1,µν vol+12

∫

MTµν

2 δg2,µν vol,

with Ta = T � Ma and δga = δg � Ma . Considering a point x ∈ M butnot on Σ we can find a vector field ξ with support contained in M1 or M2,but avoiding Σ. Thus, the condition established above applies and we haved̂ivTa = 0 on Ma −Σ, and by continuity, on Ma including Σ. Therefore, theEuler equations are still satisfied for each side of the continuous medium.

∂̂µTµν1 = 0 and ∂̂µT

µν2 = 0.

Next, let us consider a vector field ξ with compact support, extending overthe two parts M1 and M2. The Eulerian condition writes:

0=∫

M1

Tµν1 ∂̂µξνvol1+

∫

M2

Tµν2 ∂̂µξνvol2.

Integrating by parts, and after applying Euler equations, we get:∫

M1

∂̂µ�

Tµν1 ξν

�

vol1+∫

M2

∂̂µ�

Tµν2 ξν

�

vol2 = 0.

Introducing the vector Ta(ξ) defined by its components Tµa,νξν =Tµν

a ξν, withTµ

a,ν = ga,νλTλµa , we get∫

M1

d̂iv�

T1(ξ)�

vol1+∫

M2

d̂iv�

T2(ξ)�

vol2 = 0.


That is, using the identity2 d̂iv(θ)vol= d [vol(θ)] and Stoke’s theorem,

0 =∫

M1

d�

vol1�

T1(ξ)��

+∫

M2

d�

vol2�

T2(ξ)��

=∫

∂M1

vol1�

T1(ξ)�

+∫

∂M2

vol2�

T2(ξ)�

Then, we can orient M and Σ such that: ∂M1 = Σ, and then ∂M2 = −Σ.Hence,

∫

Σvol1

�

T1(ξ)�

=∫

Σvol2

�

T2(ξ)�

,

which implies and is equivalent to the condition:

vol1�

T1(δx)�

� Σ= vol2�

T2(δx)�

� Σ.

for all x ∈ Σ and δx ∈ TxΣ. Hence, the tensor T = (T1,T2) is Eulerian inpresence of the interface Σ if the Euler equations above are satisfied, togetherwith this boundary condition.Let J be a vector transverse to Σ at the point x , complete with a basisB =(e1, . . . , em) of TxΣ to get a basis (J,B) of TxM. In that basis vol1 and vol2write

Æ

|det(G1)| det andÆ

|det(G2)| det .where G1,G2 are the Gramm matrices of the metrics g1, g2. Let (J

∗,B ∗) bethe dual basis of (J,B), withB ∗ = (e∗1 , . . . , e∗m). That is, J∗J= 1, J∗ei = e∗i J=0, e∗i e j = δi j . We introduce the projectors on RJ and TxΣ, associated with thebasis (J,B) :

JJ∗ and BB ∗ =m∑

i=1

ei e∗i ,

where we identified the vectors with their matrix representations. Then, thecondition above writes, where the letters Ta denote the matrices associateswith the tensors denoted by the same letter:

J∗�

Æ

|det(G1)|T1−Æ

|det(G2)|T2

�

BB ∗ = 0. (♦)

Now, in the framework of relativity (special or general), this condition mustbe specified for each kind of medium: elastic, fluid or dust. We leave thediscussion of these cases for future work. But note already that the case ofdust, for which particles follow geodesics, leads to the special treatment thatwe shall address in the following part.

2The notation vol(θ) denotes the contraction of the form vol by the vector θ.


x

v1

v2

M

(M1,g1) (M2,g2)

Figure 1. Refraction of Geodesics

The Broken Geodesics

We consider now a curve γ in M that cuts Σ at some point x for t = 0, andlet γ1 and γ2 be the two pieces of γ,

γ1 : ]−∞, 0]→M1 and γ2 : [0,+∞[→M2, with γ1(0) = γ2(0) = x.

Let T be a tensor distribution supported by γ,

T (δg ) =12

∫ 0

−∞Tµν

1 δg1,µν dt +12

∫ +∞

0Tµν

2 δg2,µν dt .

Solving the Eulerian condition T (δL g ) = 0, by considering a vector fieldξ whose supports lie first anywhere in M1, then in M2 we get, thanks toSouriau’s computation, that γ1 and γ2 are geodesic for the respective metricand also that:

T (δg ) =k1

2

∫ 0

−∞

dxµ

dtdxν

dtδgµν dt +

k2

2

∫ +∞

0

dxµ

dtdxν

dtδgµν dt .

Where k1 and k2 are a priori two arbitrary real constants. Applied to δL gfor a compact supported vector field ξ on M, tangent to Σ on Σ, one gets, byregroupings terms, using the fact that the curves are geodesic, and integratingby parts:

0 = k1

∫ 0

−∞

dxµ

dtdxν

dt∂̂1,νξµ, dt + k2

∫ +∞

0

dxµ

dtdxν

dt∂̂2,νξµ, dt

= k1

∫ 0

−∞

∂̂1ξµ∂ xν

dxν

dtdxµ

dtdt + k2

∫ +∞

0

∂̂2ξµ∂ xν

dxν

dtdxµ

dtdt

= k1

∫ 0

−∞

d̂1ξµdt

dxµ

dtdt + k2

∫ +∞

0

d̂2ξµdt

dxµ

dtdt


= k1

∫ 0

−∞

d̂1

dt

�

ξµdxµ

dt

�

dt + k2

∫ +∞

0

d̂2

dt

�

ξµdxµ

dt

�

dt

= k1ξ(x)µ limt→0−

dxµ

dt− k2ξ(x)µ lim

t→0+

dxµ

dt.

Then, denoting by v1 and v2 the limits of the velocities

v1 = limt→0−

γ̇1(t ) and v2 = limt→0+

γ̇2(t ),

we get the Eulerian condition for the tensor distributionT at the point x ∈ Σwhich writes

k1 g1(v1,δx) = k2 g2(v2,δx) for all δx ∈Tx(Σ). (♠)

We get then a general condition which must be satisfied by the two geodesicsγ1 and γ2 at the interface, to define an Eulerian distribution. But the system isunderdetermined due to the arbitrary choice of constants k1 and k2. We candiminish the arbitrary by considering the symplectic structure of the spacesof geodesics.

Remark. — The case above admits a solution when there is no part 2, andjust the curve γ1. In that case the equation (♠) becomes g1(v1,δx) = 0 for allδx ∈Tx(Σ), that is,

v1 ⊥ Σ. (♠′)Hence, the Eulerian distributions are the geodesics which end orthogonallyat the intersection with the interface. It is the case in the Poincaré half-plan,for example. That has been observed in real world, with light rays in a salinesolution for which the concentration tends to infinity near the border.

The Symplectic Scattering

We shall use now the symplectic structure of the space of geodesics and weshall request the distribution T to be a symplectic scattering T1 → T2, inorder to lower the degree of indetermination in the equation (♠).

Let us recall now that the space Z of geodesics curves in a Riemannian (orPseudo-Riemannian) manifold (M, g ) has a canonical symplectic structure. Itis the projection of the exterior derivative dϖ of the Cartan 1-formϖ definedon Y=R× (TM−M) by

ϖ(δy) = k g (v,δx)− k2

g (v,v)δt ,


where y = (t , x,v) ∈Y and δy ∈TyY. LetZ1 andZ2 be the space of geodesicsin (M1, g1) and (M2, g2). In the two cases a parametrized geodesic that cut(transversally) the interface Σ is completely determined by (t , x,v), wherex ∈ Σ is the intersection of the geodesic with the interface, t ∈ R is the timeof the impact and v ∈TxM is the velocity of the geodesic at the point x . Thisdefined two open subsets Z?1 ⊂ Z1 and Z?2 ⊂ Z2. These two subspaces areequipped naturally with the restriction of their Cartan form ϖ1 and ϖ2, asstated above.

Because of the indetermination we have no clear map T1→T2, but a relationR that we can describe by its graph in Z?1 × Z?2. Let y1 = (t1, x1,v1) andy2 = (t2, x2,v2) represent two geodesics in each of these spaces:

y1 R y2 ⇔

t1 = t2 = t , x1 = x2 = xandk1 g1(v1,δx) = k2 g2(v2,δx) for all δx ∈TxΣ.

The first condition translates the fact that the two geodesics cut the interfaceat the same point at the same time, and the second condition describes theEulerian nature of the scattering, expressed in (♠).

Actually, we shall askR to preserve the Cartan form. It is a stronger requestbut usual in this kind of problems. Let us precise that, for the relationR , topreserve the Cartan form means that the form ϖ1ϖ2, defined on Z?1×Z?2,vanishes on R . That is, for all y ∈ R , if δy ∈ TyR then ϖ1ϖ2(δy) = 0.But

ϖ1ϖ2(δy) = k1 g1(v1,δx)−k1

2g1(v1,v1)δt

− k2 g2(v2,δx)+k2

2g2(v2,v2)δt .

Therefore,R preserves the Cartan form between the two spaces of geodesicsif and only if

k1 g1(v1,v1) = k2 g2(v2,v2). (♣)

We notice here that k1 g1(v1,v1)/2 is the moment of the symmetry group oftime translation (t , x,v) 7→ (t + e , x,v), for all e ∈ R. And preserving theCartan form is equivalent to the preservation of the energy–moment whencrossing the interface (which is not surprising). It describes a conservative— i.e. non-dissipative — process, which is the rule in symplectic mechanics.


δg

γin

γout

γ

y1 = (t1,x1,v1)

y2 = (t2,x2,v2)

Figure 2. A Metric Blob

Thus, the symplectic framework decreases the level of indetermination in thescattering process, but it remains the arbitrary of the ratio k2/k1 whichmaybelifted by a state function proper to the system considered.

We shall see that, for geometric optics, the physical context lifts indeed theindetermination.

Note 1. — This construction also answers the question of reflexion, when γ2lies in M1 too. In this case k1 = k2 and the symplectic condition states thatv1 and v2 have the same norm relatively to g1, which solves the issue withoutother considerations.

Note 2. — The requirement of symplectic scattering is justified by the follow-ing situation. Consider a manifold M, equipped with some metric g . Its spaceof parametrized geodesics G is a symplectic manifold for the pushforward ωof the differential dϖ of the Cartan form. Now consider a perturbation g ′ ofg , such that the two metrics differ only on a compact K ⊂M. For the sakeof simplicity, we assume that every geodesic for g ′ leaves the compact K inthe past and the future. The set of parametrized geodesics G ′ of (M, g ′) issymplectic for ω′. Consider a geodesic γ ∈ G ′, before entering in the com-pact K it defines a geodesic γin ∈ G , and after exiting the compact it definesa geodesic γout ∈ G , as it is described in Fig. 2. That defines a scatteringφ : γin 7→ γout on a subset of G , the geodesics crossing the compact K. Let uscheck that φ is a local symplectomorphism. Let γ ∈ G ′, and δγ and δγ ′ betwo tangents vectors at γ ∈ G . Let y = (t , x,v) be an initial condition forγ, let δy = (δt ,δx,δv) and δ′y = (δ′t ,δ′x,δ′v) be two tangent vectors pro-jecting on δγ and δ′γ. Then, ω′(δγ,δγ ′) = dϖ ′(δy,δ′y). But we can evaluatedϖ ′(δy,δ′y) at a point y1 = (t1, x1,v1) before γ entering the compact K. Inthis case ω′(δγ,δγ ′) = dϖ ′(δy1,δ

′y1) = dϖ(δy1,δ′y1) = ω(δγin,δ′γin). And

we can also evaluate dϖ ′(δy,δ′y) at a point y2 = (t2, x2,v2) after γ leaving


the compact K. in this case ω′(δγ,δγ ′) = dϖ ′(δy2,δ′y2) = dϖ(δy2,δ

′y2) =ω(δγout,δ

′γout). Therefore, ω(δγout,δ′γout) = ω(δγin,δ′γin), that is φ

∗(ω) = ω.The scattering φ : γin 7→ γout, which is bijective (by t 7→ −t ), is then a symplec-tomorphism. Hence, assuming that the scattering of geodesics by an interface— dividing two different media — is symplectic, is completely legitimate.

The Case of Light

Let us consider now the special case of light rays propagating in two homoge-neous media with different indices n1 and n2, separated by the hypersurfaceΣ. We shall see that the particular context of geometric optics will achieveto lift the indetermination in the rule (♠), with the condition of symplecticscattering (♣). Let v1 and v2 be the speed of light in the two different media.Then,

g1 = n21 g0 and g2 = n2

2 g0 with n1 =cv1

and n2 =cv2

,

where g0 is an ambient smooth metric on M, the vacuum metric. AssumingM⊂ R3, we chose g0 to be the standard scalar product, and we note as usualg0(u,v) = u · v and g0(v,v) = ‖v‖2. Let us recall that we defined the colorcoefficients as

k1 =hν1

c2and k2 =

hν2

c2,

and let,

v1 = v1u1 and v2 = v1u2 with ‖u1‖2 = ‖u2‖

2 = 1.

Next, the symplectic condition (♣) writes

k1n21 ‖v1‖

2 = k2n22 ‖v2‖

2, that is, k1c2

v21

v21 = k2

c2

v22

v22 .

Hence,k1 = k2 i.e. ν1 = ν2.

The frequency of the light ray, its color, does not change when crossing theinterface. Now, the covariant condition (♠) writes, for all δx ∈TxΣ,

n21 v1·δx = n2

2 v2·δx ⇔ c2

v21

v1u1·δx =c2

v22

v2u2·δx ⇒ cv1

u1·δx =cv2

u2·δx.

That is, (n1u1− n2u2) · δx = 0 for all δx ∈TxΣ. And then,

n2u2 = n1u1+αn,


where n is the unit normal vector to Σ at x , and α some number. By decom-posing the unit vectors u1 and u2 in an orthonormal basis completing n — inthe plane spanned by u1 and n — and denoting by θi the angles between theui and the normal n, we deduce the classical Snell–Descartes law of refractionof light,

n1 sin(θ1) = n2 sin(θ2). (♥)

In Conclusion, the law of refraction of light becomes a consequence of thegeneral covariance principle, conjugated with the symplectic structure of thespace of rays. The case of reflexion is also covered and gives as usual sin(θ1) =sin(θ2).It will be noted again that in these cases, the distribution tensor T defines ascattering map T1→T2, from the inward geodesic γ1 to the outward geodesicγ2.

Behind the Scenes

It is very intriguing that the evolution of material media, that are generallydescribed by a principle of least action, can be also described by a principleof invariance (actually covariance) by diffeomorphisms. The medium can beelastic, or a particle, or even a light ray refracted by an interface.

There is always some kind of mystery behind the principle of least action,sort of God willing principle. Or the Nature which — by a hidden will —decides at each instant what is the best for the system in its immediate future.Every student who has been there once, knows this feeling of some magic thatgoverns the world of matter.

And suddenly, we have the same behavior described just by an invariance prin-ciple, where all the magic has disappeared: thatmaterial presence and its evolu-tion do not depend on the way they are described, they have an objective nature.And it is this objectivity which is captured by this principle of general covari-ance: the laws of nature are described by a covector on the space of geometriesmodulo diffeomorphisms. How is that possible?

Actually, these two principles are not as independent as they seem. The princi-ple of least action and the principle of general covariance are indeed the two facesof the same coin. And to bring that to light, we shall consider the followingtoy model.

Let C be a space, we shall call the space of fields. LetM be a space, we shallcall the space of geometries. Here C andM will be manifolds. Let G be a Lie


group acting on the fields and the geometries. Let c ∈C , g ∈M and φ ∈G.Let φ∗(c) and φ∗(g ) be the action of φ on c and g . Assume that the actionof G on C is transitive. LetA : C ×M → R be a smooth map we call theaction function. Assume that the action is invariant under the diagonal actionof G on C ×M . That is

A (φ∗(c),φ∗(g )) =A (c , g ) for all φ ∈G.

Next, consider a 1-parameter subgroup s 7→ φs in G and let us put

δLc =∂ φs∗(c)∂ s

�

�

�

�

s=0and δL g =

∂ φ∗s (g )∂ s

�

�

�

�

�

s=0

,

where φ∗ = φ−1∗. Therefore, taking the derivative of the invariance identity

of the action above, we get that for all infinitesimal action δL of the group Gwe have

J (δLc)−T (δL g ) = 0 with J = ∂A∂ c

and T = ∂A∂ g

.

Thus, obviously, if T (δL g ) = 0, then J (δLc), and conversely. Hence:

(1) If c is a critical point forAg : c 7→A (c , g ), then T (δL g ) = 0 for allδL g . That is, T is Eulerian.

(2) IfT is Eulerian, thenJ (δLc) = 0 for all δLc . But sinceG is transitiveon C , we get J (δc) = 0 for any variation δc . Which means that

δAg =∂Ag (cs )

∂ s

�

�

�

�

�

s=0

= 0 with Ag : c 7→A (c , g ),

for all smooth path s 7→ cs . Therefore, c is a critical point of the actionAg , and a solution of the least action principle for this action.

Of course, for the cases treated in general the space of fields and the space ofgeometries are not manifolds but infinite dimensional spaces. There is cer-tainly a right way to treat these situations rigorously, from the point of viewof diffeology for example. But that is not the point here. The idea here isjust to see how the principle of general covariance withdraws the magic fromthe principle of least action, by giving the main role to the laws of symme-tries and the principle of objectivity/relativity which definitely deserve it. Atthe same time the principle of general covariance is a well founded a priorigeneralization of the least action principle.


References

[Sou74] Jean-Marie Souriau. Modèle de particule à spin dans le champ électromagnétique etgravitationnel. Ann. Inst. Henri Poincaré, XX A, 1974.

[Ste06] Shlomo Sternberg. General Covariance and the Passive Equations of Physics. AlbertEinstein Memorial Lecture, ISBN 965-208-173-6. The Israel Academy of Sciences andHumanities, Jerusalem 2006.

Patrick Iglesias-Zemmour — Aix Marseille Univ, CNRS, Centrale Marseille, I2M,

Marseille, France — & — Einstein Institute of Mathematics Edmond J. Safra Cam-

pus, The Hebrew University of Jerusalem Givat Ram. Jerusalem, 9190401, Israel.

E-mail address: [email protected]

refraction and reflexion according to the principle...

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